Congressional Republicans rushed late Friday to develop a new plan for reopening the government and avoiding a first-ever default in hopes of crafting a strategy that can win the support of the White House before financial markets open Monday.

Talks on Capitol Hill advanced with a new urgency after President Obama rejected House Speaker John A. Boehner’s (R) offer to raise the debt limit through late November to give the parties time to negotiate a broader budget deal.

Briefing reporters after financial markets closed for the week, White House press secretary Jay Carney praised a “new willingness” among Republicans to end the government shutdown — now in its 12th day — and to acknowledge that default on the national debt “would be catastrophically damaging.”

But with the Treasury Department due to exhaust its borrowing authority in just six days, Carney said the president would not agree to go through another round of economy-rattling talks in six weeks, just before the Christmas shopping season.

“It at least looks as if there’s a possibility of making some progress here,” Carney said. But, “we have to remove these demands for leverage essentially using the American people and the American economy.”

Before Carney spoke, Obama telephoned Boehner and the two men agreed to keep talking, aides said. Afterward, GOP senators marched into Boehner’s office and counseled him to adopt an approach they had presented to Obama earlier in the day, during their own meeting at the White House.

With Republicans getting battered in public opinion polls over the shutdown, Senate GOP leaders urged Boehner to join them in supporting a single, big-bang measure that would open the government and raise the debt limit in one fell swoop.

“I laid out some of those ideas, and the question is, can the House find a center of gravity to open the government up around those ideas,” Sen. Lindsey Graham (R-S.C.) said after exiting the speaker’s office with Sen. Saxby Chambliss (R-Ga.). The two men, former House members, have been close friends with Boehner for almost 20 years.

Details were still fluid late Friday, but the latest 23-page draft of the emerging measure would immediately end the shutdown and fund federal agencies for six months at current spending levels. It would maintain deep automatic cuts known as the sequester, but give agency officials flexibility to decide where the cuts should fall.

In addition, the proposal would also raise the debt limit through Jan. 31, 2014. Lawmakers were considering whether to include a provision that would direct the House and Senate Budget committees to immediately enter negotiations over broader budget issues and to issue a report by Jan. 15, 2014. If an agreement could be reached, it would clear a path for another increase in the debt limit later that month, without additional drama.

In exchange, Republicans were seeking what they called a few “fig leafs” — minor adjustments to Obama’s new health-care initiative. The first would delay for two years a 2.3-percent tax on medical devices that is unpopular in both parties. The second would require internal auditors to ensure that people who get tax subsidies to buy health insurance are in fact eligible.

Another option under consideration but not included in the latest draft would reduce the number of workers required to receive health coverage from an employer, by changing the definition of a full-time worker from 30 hours a week to 40 hours a week.

In an interview with a Kentucky newspaper Friday, Senate Minority Leader Mitch McConnell (R-Ky.) signaled that he was helping to shepherd the effort to reach a compromise with Democrats.

“We’re in one of those situations right now where it’s going to require some sort of coming together here to get past the current impasse. And I’m going to continue to work on that,” McConnell told the Herald-Leader of Lexington.

By late Friday, talks over the measure were proceeding on multiple tracks. In the Senate, negotiations had advanced far enough that Senate Republicans — led by Sens. Susan Collins (Maine), Kelly Ayotte (N.H.) and Lisa Murkowski (Alaska) — sent draft language to their Democratic counterparts, including Sen. Charles Schumer (D-N.Y.), a trusted ally of Senate Majority Leader Harry Reid (D-Nev.).

Meanwhile, Boehner was huddled with his top lieutenants in a hideaway on the first floor of the Capitol, reviewing his options over Chinese carryout and cigarettes.

The Senate proposal differs in critical ways from the approach Boehner sold earlier this week to his rank and file, who had insisted on using the threat of the shutdown to try to undermine the Affordable Care Act.

Given those demands, Boehner had offered to lift the debt limit for six weeks to clear space for negotiations over overhauling the tax code, trimming federal entitlement spending and reforming the health care law. The government would remain shuttered unless Obama agreed to those talks.

House Republicans also proposed to roll back a portion of the sequester cuts, a top Democratic priority. But those cuts would have to be replaced with equal reductions to Medicare spending, such as Obama’s proposal to make well-off seniors pay more for coverage.

It was unclear how long the House offer would have kept the government open without further negotiations. But House Appropriations Chairman Harold Rogers (R-Ky.) quickly blasted the Senate plan to extend temporary funding for six months, calling it “disastrous.”

“It is a punt to the executive branch for the Congress not to exercise judgement about where money is spent,” Rogers said in a statement.

Boehner has scheduled a meeting of the entire GOP conference for 9 a.m. Saturday, but it was unclear whether he would present details of the proposal emerging in the Senate. Senate Republicans were hopeful that Boehner could build support for the plan and push it through the House first.

That could not only help get it to Obama’s desk faster, but also preserve Boehner’s political standing by avoiding a repeat of the New Year’s Day deal over the so-called “fiscal cliff.”

Then, a Senate-passed measure to stop scheduled tax hikes for the vast majority of Americans won the support of only 85 Republicans in the House. More than 150 GOP lawmakers opposed the bill, including some of Boehner’s top deputies, and the humiliating loss emboldened conservatives to vote against his re-election as speaker.

Senate Republicans cautioned, however, that Boehner doesn’t have much time to work things out. GOP senators are eager not only to get the government back to work but to raise the $16.7 trillion debt limit before Thursday, when the Treasury Department has said it would exhaust its ability to conserve cash. Without an increase in the debt limit, independent analysts say the Treasury would begin missing payments by Nov. 1.

GOP senators and aides said Boehner has been told that Senate Republicans will negotiate their own pact with Senate Democrats if he fails to act.

“From my point of view, it’d be better for the country if the House led,” Graham said. “It’s important that we continue to talk among ourselves as senators,” he added, but “if it came out of the House, it would be better.”

Even as negotiations progressed rapidly in the Senate, the uncertainty over procedure cast a shadow of doubt over their ultimate success.

“It’s encouraging to me that the president is now negotiating with both the House and the Senate, after saying that he wouldn’t,” Collins said. “But it is very uncertain to me what the outcome is going to be.”

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